Log concavity of the maximum of dependent Gaussians

Question

Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in general?

Answer 1

$\newcommand{\R}{\mathbb{R}} \renewcommand{\P}{\operatorname{\mathsf P}} \newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}}$

This is false in general. E.g., let $Z_1=U$ and $Z_2=|V|\,\text{sign}\,U$, where $U,V$ are iid standard normal random variables. Let $F$ be the cdf of $\max(Z_1,Z_2)$ and $L:=\ln F$. Let also $\Phi$ denote the standard normal cdf. Then $Z_1$ and $Z_2$ are each Gaussian and for $x\ge0$ \begin{align*} F(x)&=\P(U<x,|V|\,\text{sign}\,U<x) \\ &=\P(U<0)\P(|V|>-x)+\P(0<U<x)\P(|V|<x) \\ &=\tfrac12+[\Phi(x)-\Phi(0)][\Phi(x)-\Phi(-x)], \end{align*} which yields $L''(0+)=4/\pi>0$, so that $F$ is not log concave.

However, if $Z_1,\dots,Z_n$ are jointly normal, then for $x\in\R$ \begin{equation} F(x)=\int_{\R^n}f(z_1,\dots,z_n)\ii{z_1<x}\cdots\ii{z_n<x}\,dz_1\cdots dz_n, \end{equation} where $F$ is the cdf of $\max(Z_1,\dots,Z_n)$, $f$ is the pdf of $(Z_1,\dots,Z_n)$, and $\ii\cdot$ denotes the indicator. So, by the Prékopa–Leindler theorem (see Section Applications in probability and statistics), $F$ is log concave -- since $f$ is log concave, $\ii{z_i<x}$ is log concave in $(z_i,x)$ for each $i$, and the product of log-concave functions is log concave.